Outline for CAD conference submission
Introduction
* Counting parameters
* interpolating spline
* variational definitions - notion of optimality
* Extensionality (follows from variational def)
Key result: two parameters + extensional -> cut from rigid curve
So problem reduces to finding best curve.
Counting parameters
* Curve segments in splines are potentially from infinite dimensional
space
* In practice, most splines use curves drawn from finite-dimensional
manifold
* Subtract out translation, rotation, uniform scaling (i.e. align to
endpoints of chord)
* Observation: dimensionality never > 2 * (num control pts - 2), and
most splines are defined as a composition
Concrete representation of parameter space is tangent at endpoints.
Defn of 2 parameter: segment is uniquely determined by tangents at
endpoints. (maybe need to
Catalog of existing splines that fit into 2 parameter framework:
* Euler spiral spline (Mehlum Kurgla 2)
* MEC (Birkhoff, etc etc)
* IKARUS
* Hobby
* Circle spline
Observation: 2-parameter is sufficient for G2 continuity, without
cheating.
Variational defs and extensionality
* Cite Knuth for extensionality def'n.
* MEC is defined by minimizing L2 norm of curvature.
* Any spline which is defined variationally is by construction
extensional. Argument by adding point - if curve is different it
would have had lower fnl, and that could have served in first place.
2 parameters + extensional -> cut from rigid curve
Start by giving examples: MEC, Euler spiral. Also parameter-counting
argument. s0 and s1 th0 and th1.
Give argument in terms of extending the curve (figure
two_continue.pdf). This needs carefully crafted assumptions as far as
uniqueness and existence of curve segment given th0, th1.
Properties of curve properties of spline
Just about any curve can be used. Some (Euler spiral) have unique
s0,s1 soln for given th0,th1; others (MEC) need disambiguating
rule. Properties of the resulting spline follow from properties of the
generating curve.
* Inflection point (odd symmetry) -> spline can contain inflection pt.
* Monotonic curvature monotonic curvature ("salience")
+ might mention monotonic curvature -> no self-intersection within
segment. Or is this a digression?
* lim{n->\infty} k'/k = 0 -> roundness
* k' near 0 -> locality. Less means better locality. This is
empirical.
Aesthetic curves and user study
* Aesthetic curve family is good candidate for our generating curve -
it has monotonic curvature, shallow k' near 0 (for higher
exponents), and a good rationale that the eye is more sensitive to
k' in low k regions.
* Present user study results.
* Result that \alpha = 1.5 is preferred proves that MEC is not
accurate predictor of user choice. (note, optimizing MEC also
doesn't give roundness)
Conclusion
This is a pretty good approach to the best possible spline. G2
continuity is enough (for human visual perception, anyway), and
2-parameter is sufficient to get that. Any reasonable definition of
"best" will imply extensionality (certainly any1 definition in terms of
variational principle implies it formally).
We don't need a variational formulation to arrive at a "best"
spline. Can do it empirically by choosing the most pleasing curve.
(good opportunity to recap user study).
Two-parameter is a useful taxonomic organizing principle. It covers a
range of existing important splines.